Let $\alpha$ and $\beta$ be the roots of equation $x ^ { 2 } - 6 x - 2 = 0$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n } , \forall n \geq 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } }$ is equal to (1) - 3 (2) 6 (3) - 6 (4) 3
A complex number $z$ is said to be unimodular if $| z | = 1$. Let $z _ { 1 }$ and $z _ { 2 }$ are complex numbers such that $\frac { z _ { 1 } - 2 z _ { 2 } } { 2 - z _ { 1 } \bar { z } _ { 2 } }$ is unimodular and $z _ { 2 }$ is not unimodular, then the point $z _ { 1 }$ lies on a (1) circle of radius $\sqrt { 2 }$ (2) straight line parallel to $x$-axis (3) straight line parallel to $y$-axis (4) circle of radius 2
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $( 0,0 ) , ( 0,41 )$ and $( 41,0 )$ is (1) 780 (2) 901 (3) 861 (4) 820
Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$, each having at least three elements is (1) 510 (2) 219 (3) 256 (4) 275
If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ are three geometric means between $l$ and $n$, then $G _ { 1 } ^ { 4 } + 2 G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 }$ equals (1) $4 l ^ { 2 } m ^ { 2 } n ^ { 2 }$ (2) $4 l ^ { 2 } m n$ (3) $4 l m ^ { 2 } n$ (4) $4 l m n ^ { 2 }$
Locus of the image of the point $( 2,3 )$ in the line $( 2 x - 3 y + 4 ) + k ( x - 2 y + 3 ) = 0 , k \in \mathbb{R}$, is a (1) Circle of radius $\sqrt { 3 }$ (2) Straight line parallel to $x$-axis. (3) Straight line parallel to $y$-axis. (4) Circle of radius $\sqrt { 2 }$
The number of common tangents to the circles $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 18 y + 26 = 0$, is (1) 4 (2) 1 (3) 2 (4) 3
Let $O$ be the vertex and $Q$ be any point on the parabola, $x ^ { 2 } = 8 y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1 : 3$, then the locus of $P$ is (1) $x ^ { 2 } = 2 y$ (2) $x ^ { 2 } = y$ (3) $y ^ { 2 } = x$ (4) $y ^ { 2 } = 2 x$
The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$, is (1) 27 (2) $\frac { 27 } { 4 }$ (3) 18 (4) $\frac { 27 } { 2 }$
The mean of a data set comprising of 16 observations is 16. If one of the observation value 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is (1) 14.0 (2) 16.8 (3) 16.0 (4) 15.8
If the angles of elevation of the top of a tower from three collinear points $A , B$ and $C$ on a line leading to the foot of the tower are $30 ^ { \circ } , 45 ^ { \circ }$ and $60 ^ { \circ }$ respectively, then the ratio $AB : BC$, is (1) $2 : 3$ (2) $\sqrt { 3 } : 1$ (3) $\sqrt { 3 } : \sqrt { 2 }$ (4) $1 : \sqrt { 3 }$
$A = \left[ \begin{array} { c c c } 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array} \right]$ is a matrix satisfying the equation $A A ^ { T } = 9 I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $( a , b )$ is equal to (1) $( - 2 , - 1 )$ (2) $( 2 , - 1 )$ (3) $( - 2,1 )$ (4) $( 2,1 )$
The set of all values of $\lambda$ for which the system of linear equations: $2 x _ { 1 } - 2 x _ { 2 } + x _ { 3 } = \lambda x _ { 1 }$ $2 x _ { 1 } - 3 x _ { 2 } + 2 x _ { 3 } = \lambda x _ { 2 }$ $- x _ { 1 } + 2 x _ { 2 } = \lambda x _ { 3 }$ has a non-trivial solution, (1) Contains more than two elements. (2) Is an empty set. (3) Is a singleton. (4) Contains two elements.
If the function $g ( x ) = \left\{ \begin{array} { c c } k \sqrt { x + 1 } , & 0 \leq x \leq 3 \\ m x + 2 , & 3 < x \leq 5 \end{array} \right.$ is differentiable, then the value of $k + m$ is (1) 4 (2) 2 (3) $\frac { 16 } { 5 }$ (4) $\frac { 10 } { 3 }$
The normal to the curve $x ^ { 2 } + 2 x y - 3 y ^ { 2 } = 0$, at $( 1,1 )$ (1) Meets the curve again in the fourth quadrant (2) Does not meet the curve again (3) Meets the curve again in the second quadrant (4) Meets the curve again in the third quadrant
Let $f ( x )$ be a polynomial of degree four and having its extreme values at $x = 1$ and $x = 2$. If $\lim _ { x \rightarrow 0 } \left[ 1 + \frac { f ( x ) } { x ^ { 2 } } \right] = 3$, then $f ( 2 )$ is equal to (1) 4 (2) - 8 (3) - 4 (4) 0
The area (in sq. units) of the region described by $\left\{ ( x , y ) : y ^ { 2 } \leq 2 x \text{ and } y \geq 4 x - 1 \right\}$ is (1) $\frac { 9 } { 32 }$ sq. units (2) $\frac { 7 } { 32 }$ sq. units (3) $\frac { 5 } { 64 }$ sq. units (4) $\frac { 15 } { 64 }$ sq. units
Q86
First order differential equations (integrating factor)View
Let $y ( x )$ be the solution of the differential equation $( x \log x ) \frac { d y } { d x } + y = 2 x \log x , ( x \geq 1 )$. Then $y ( e )$ is equal to (1) $2 e$ (2) $e$ (3) 0 (4) 2
Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that no two of them are collinear and $( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }$. If $\theta$ is the angle between vectors $\vec { b }$ and $\vec { c }$, then a value of $\sin \theta$ is (1) $\frac { - 2 \sqrt { 3 } } { 3 }$ (2) $\frac { 2 \sqrt { 2 } } { 3 }$ (3) $\frac { - \sqrt { 2 } } { 3 }$ (4) $\frac { 2 } { 3 }$
The distance of the point $( 1,0,2 )$ from the point of intersection of the line $\frac { x - 2 } { 3 } = \frac { y + 1 } { 4 } = \frac { z - 2 } { 12 }$ and the plane $x - y + z = 16$, is (1) 13 (2) $2 \sqrt { 14 }$ (3) 8 (4) $3 \sqrt { 21 }$
The equation of the plane containing the line of intersection of $2 x - 5 y + z = 3 ; x + y + 4 z = 5$, and parallel to the plane, $x + 3 y + 6 z = 1$, is (1) $2 x + 6 y + 12 z = - 13$ (2) $2 x + 6 y + 12 z = 13$ (3) $x + 3 y + 6 z = - 7$ (4) $x + 3 y + 6 z = 7$